A006784 -id:A006784 - cp's OEIS frontend

A028257 Engel expansion of sqrt(3).

1, 2, 3, 3, 6, 17, 23, 25, 27, 73, 84, 201, 750, 24981, 46882, 119318, 121154, 242807, 276226, 3009377, 3383197, 37871208, 45930966, 261728403, 281868388, 3021299588, 3230725884, 13646315477, 30951797814, 80602040381, 1016719946612, 49448385811777

Offset: 1

Naoki Sato (naoki(AT)math.toronto.edu)

nonn

T. D. Noe, Table of n, a(n) for n = 1..300
Eric Weisstein's World of Mathematics, Engel Expansion
Index entries for sequences related to Egyptian fractions

Cf. A006784 (for definition of Engel expansion), A220335.

EngelExp[A_,n_]:=Join[Array[1&,Floor[A]],First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]],Expand[ #[[1]]#[[2]]-1]}&,{Ceiling[1/(A-Floor[A])],A-Floor[A]},n-1]]; EngelExp[N[3^(1/2),7! ],50] (* Vladimir Joseph Stephan Orlovsky, Jun 08 2009 *)

For a number x (here sqrt(3)), define a(1) <= a(2) <= a(3) <= ... so that x = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + ... by x(1) = x, a(n) = ceiling(1/x(n)), x(n+1) = x(n)*a(n) - 1.

Better name and more terms from Simon Plouffe

More terms from Sean A. Irvine, Dec 16 2019

A054543 Engel series expansion (or "Egyptian product") for Catalan's constant G.

Original entry on oeis.org

2, 2, 2, 4, 4, 5, 5, 12, 13, 41, 110, 172, 248, 309, 3146, 5919, 21959, 22299, 30892, 401838, 1719239, 30576561, 262313756, 630913752, 3242181301, 3250783944, 13827502849, 40152067840, 137791590233, 2514510232695, 3217773878849

Offset: 1

Jeppe Stig Nielsen, Apr 09 2000

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 53-59.

Simon Plouffe, Table of n, a(n) for n = 1..212
Steven R. Finch, Catalan's Constant [Broken link]
Steven R. Finch, Catalan's Constant [From the Wayback machine]
Oleg Marichev, Jonathan Sondow, and Eric W. Weisstein, Catalan's Constant, MathWorld.
Eric Weisstein's World of Mathematics, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A006784, A028254, A028257, A006752, A104338, A014538, A153069, A153070, A118323.

EngelExp[A_,n_]:=Join[Array[1&,Floor[A]],First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]],Expand[ #[[1]]#[[2]]-1]}&,{Ceiling[1/(A-Floor[A])],A-Floor[A]},n-1]]; EngelExp[N[Catalan,7! ],50] (* Vladimir Joseph Stephan Orlovsky, Jun 08 2009 *)

A232325 Engel expansion of 1 to the base Pi.

Original entry on oeis.org

4, 12, 72, 2111, 14265, 70424, 308832, 4371476, 320218450, 1101000257, 14020589841, 102772320834, 963205851651, 5997003656523, 50649135127796, 640772902021920, 2101002284323870, 35029677728070645, 176996397541889098, 1433436623499128186

Offset: 0

Peter Bala, Nov 25 2013

Let r and b be positive real numbers. We define an Engel expansion of r to the base b to be a (possibly infinite) nondecreasing sequence of positive integers [a(0), a(1), a(2), ...] such that we have the series representation r = b/a(0) + b^2/(a(0)*a(1)) + b^3/(a(0)*a(1)*a(2)) + .... Depending on the values of r and b such an expansion may not exist, and if it does exist it may not be unique.
When b = 1 we recover the ordinary Engel expansion of r. See A181565 and A230601 for some predictable Engel expansions to a base b other than 1.
In the particular case that the base b >= 1 and 0 < r < b then we can find an Engel expansion of r to the base b using the following algorithm:
Choose values for r and b.
Define the map f(x) (which depends on the base b) by f(x) = x/b*ceiling(b/x) - 1 and let f^(n)(x) denote the n-th iterate of the map f(x), with the convention that f^(0)(x) = x.
For n = 0, 1, 2, ... define the integer a(n) = ceiling(b/f^(n)(r)) until f^n(r) = 0.
When b >= 1 and 0 < r < b the sequence a(n) produced by this algorithm provides an Engel expansion of r to the base b.
For the present sequence we apply this algorithm with r := 1 and with the base b := Pi.
We can also get an alternating series representation for r in powers of b (still assuming b >= 1 and 0 < r < b), called a Pierce series expansion of r to the base b, by running the above algorithm but now with input values -r and base b. See A232326.
In addition, we can obtain two further series expansions for r in powers of b by running the algorithm with either the input values r and base -b or with the input values -r and base -b. See examples below. See A232327 and A232328 for other examples of these types of expansions.

			Truncation F_5(z) = 1 - ( z/4 + z^2/(4*12) + z^3/(4*12*72) + z^4/(4*12*72*2111) + z^5/(4*12*72*2111*14265) ). The polynomial has a positive real zero at z = 3.14159 26535 (9...), which agrees with Pi to 10 decimal places.
Comparison of generalized Engel expansions of 1 to the base Pi.
A232325: Engel series expansion of 1 to the base Pi
1 = Pi/4 + Pi^2/(4*12) + Pi^3/(4*12*72) + Pi^4/(4*12*72*2111) + ....
A232326: Pierce series expansion of 1 to the base Pi
1 = Pi/3 - Pi^2/(3*69) + Pi^3/(3*69*310) - Pi^4/(3*69*310*1017) + - ....
Running the algorithm with the input values r = 1 and base -Pi produces the expansion
1 = Pi/3 - Pi^2/(3*70) - Pi^3/(3*70*740) + Pi^4/(3*70*740*6920) + - - + ....
Running the algorithm with the input values r = -1 and base -Pi produces the expansion
1 = Pi/4 + Pi^2/(4*11) - Pi^3/(4*11*73) - Pi^4/(4*11*73*560) + + - - ....

Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion

Cf. A014014, A006784, A061233, A185565, A230601, A232326, A232327, A232328, A303877.

# Define the n-th iterate of the map f(x) = x/b*ceiling(b/x) - 1
map_iterate := proc(n,b,x) option remember;
if n = 0 then
   x
else
  -1 + 1/b*thisproc(n-1,b,x)*ceil(b/thisproc(n-1,b,x))
end if
end proc:
# Define the terms of the expansion of x to the base b
a := n -> ceil(evalf(b/map_iterate(n,b,x))):
Digits:= 500:
# Choose values for x and b
x := 1: b:= Pi:
seq(a(n), n = 0..19);

a(n) = ceiling(Pi/f^(n)(1)), where f^(n)(x) denotes the n-th iterate of the map f(x) = x/Pi*(ceiling(Pi/x)) - 1, with the convention that f^(0)(x) = x.
Engel series expansion of 1 to the base Pi:
1 = Pi/4 + Pi^2/(4*12) + Pi^3/(4*12*72) + Pi^4/(4*12*72*2111) + ....
The associated power series F(z) := 1 - ( z/4 + z^2/(4*12) + z^3/(4*12*72) + z^4/(4*12*72*2111) + ...) has a zero at z = Pi. Truncating the series F(z) to n terms produces a polynomial F_n(z) with rational coefficients which has a real zero close to Pi. See below for an example.

A028254 Engel expansion of sqrt(2).

Original entry on oeis.org

1, 3, 5, 5, 16, 18, 78, 102, 120, 144, 251, 363, 1402, 31169, 88630, 184655, 259252, 298770, 4196070, 38538874, 616984563, 1975413035, 5345718057, 27843871197, 54516286513, 334398528974, 445879679626, 495957494386, 2450869042061, 2629541150529, 4088114099885

Offset: 1

Naoki Sato (naoki(AT)math.toronto.edu)

nonn

For a number x (here sqrt(2)), define a(1) <= a(2) <= a(3) <= ... so that x = 1/a(1) + 1/a(1)a(2) + 1/a(1)a(2)a(3) + ... by x(1) = x, a(n) = ceiling(1/x(n)), x(n+1) = x(n)a(n) - 1.

			sqrt(2) = 1.4142135623730950488...
1 + 1/3 = 4/3 = 1.3333333333333333333...; sqrt(2) - 4/3 = 0.080880229...
1 + 1/3 + 1/15 = 7/5 = 1.4; sqrt(2) - 7/5 = 0.014213562373...
1 + 1/3 + 1/15 + 1/75 = 106/75 = 1.4133333333333333...; sqrt(2) - 106/75 = 0.000880229...

T. D. Noe, Table of n, a(n) for n = 1..300
Benoît Rittaud, La porte d’harmonie, Images des Mathématiques, CNRS, 2009 (in French).
Naoki Sato, Home page (broken link)
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Pythagoras's Constant

Cf. A002193 (decimal expansion), A006784 (for definition of Engel expansion), A028257 (Engel expansion of sqrt(3)).

expandEngel[A_, n_] := Join[Array[1 &, Floor[A]], First @ Transpose @ NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; expandEngel[N[2^(1/2), 7!], 47] (* Vladimir Joseph Stephan Orlovsky, Jun 08 2009 *)

More terms from Simon Plouffe, Jan 05 2002

A068377 Engel expansion of sinh(1).

Original entry on oeis.org

1, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8190

Offset: 1

Benoit Cloitre, Mar 03 2002

This sequence is also the Pierce expansion of sin(1). - G. C. Greubel, Nov 14 2016

Simon Plouffe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Engel Expansion.
Eric Weisstein's World of Mathematics, Hyperbolic Sine.
Eric Weisstein's World of Mathematics, Pierce Expansion.
Wikipedia, Engel Expansion.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A002943, A006784, A068379, A068380, A073742 (sinh(1)).

Join[{1}, Table[(2 n - 2) (2 n - 1), {n, 2, 50}]] (* Bruno Berselli, Aug 04 2015 *)
LinearRecurrence[{3,-3,1}, {1,6,20,42}, 25] (* G. C. Greubel, Oct 27 2016; a(1)=1 by Georg Fischer, Apr 02 2019*)
Rest@ CoefficientList[Series[x (1 + 3 x + 5 x^2 - x^3)/(1 - x)^3, {x, 0, 46}], x] (* Michael De Vlieger, Oct 28 2016 *)
PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Sin[1] , 7!], 50] (* G. C. Greubel, Nov 14 2016 *)

A068377(n)=(n+n--)*n*2+!n \\ M. F. Hasler, Jul 19 2015

A068377 = lambda n: rising_factorial(n*2,2) if n>0 else 1
print([A068377(n) for n in (0..45)]) # Peter Luschny, Aug 04 2015

a(n) = (2*n-2)*(2*n-1) = A002943(n-1) = 2*A000217(2n-2) for n>1. [Corrected and extended by M. F. Hasler, Jul 19 2015]
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
G.f.: x*(1 + 3*x + 5*x^2 - x^3)/(1-x)^3. (End)
E.g.f.: -2 + x + 2*(1 - x + 2*x^2)*exp(x). - G. C. Greubel, Oct 27 2016
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = 2 - log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - Pi/4 - log(2)/2. (End)

A195601 Engel expansion of beta = 3/(2*log(alpha/2)); alpha = A195596.

Original entry on oeis.org

1, 2, 2, 2, 2, 5, 5, 5, 20, 36, 78, 842, 5291, 10373, 17340, 28619, 35586, 93572, 98045, 2470364, 13603654, 14328528, 16490766, 833971648, 1788088151, 9592330101, 10952282168, 40005288076, 54302548920, 118523737357, 776601533408, 1241894797770, 24485470725324

Offset: 1

Alois P. Heinz, Sep 21 2011

nonn

beta = 1.95302570335815413945406288542575380414251340201036319609354... is used to measure the expected height of random binary search trees.

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
B. Reed, The height of a random binary search tree, J. ACM, 50 (2003), 306-332.
Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A195599 (decimal expansion), A195600 (continued fraction), A195581, A195582, A195583, A195596, A195597, A195598.

alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
beta:= 3/(2*log(alpha/2)):
engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
Digits:=400: engel(evalf(beta), 39);

f:= N[-1/ProductLog[-1/(2*E)], 500001]; EngelExp[A_, n_]:= Join[Array[1 &, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; EngelExp[N[3/(2*Log[f/2]), 500000], 25] (* G. C. Greubel, Oct 21 2016 *)

beta = 3/(2*log(alpha/2)) = 3*alpha/(2*alpha-2), where alpha = A195596 = -1/W(-exp(-1)/2) and W is the Lambert W function.

A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1).

A195598 Engel expansion of alpha, the unique solution on [2,oo) of the equation alphalog((2e)/alpha)=1.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 5, 10, 15, 18, 102, 114, 246, 394, 1051, 3044, 50263, 111686, 128162, 273256, 583069, 927699, 7299350, 10833746, 15187876, 67314562, 2141820499, 4969978969, 10131201410, 49316153957, 221808008142, 275241196373, 1466049587038, 3406190692970

Offset: 1

Alois P. Heinz, Sep 21 2011

nonn

alpha = 4.31107040700100503504707609644689027839156299804028805066937... is used to measure the expected height of random binary search trees.

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
B. Reed, The height of a random binary search tree, J. ACM, 50 (2003), 306-332.
Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A195596 (decimal expansion), A195597 (continued fraction), A195581, A195582, A195583, A195599, A195600, A195601.

alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
Digits:=400: engel(evalf(alpha), 39);

alpha = -1/W(-exp(-1)/2), where W is the Lambert W function.

A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1), with beta = 1.953... (A195599).

A220393 A modified Engel expansion of Pi.

Original entry on oeis.org

1, 1, 1, 8, 14, 2, 2, 3, 4, 5, 96, 115, 8, 2, 2, 2, 81, 160, 2, 6, 355, 140, 4, 12, 6, 2, 2, 3, 4, 3, 46, 66, 4, 2, 9, 16, 3, 4, 3, 4, 2, 2, 4, 9, 4, 2, 4, 33, 20, 2, 3, 4, 2, 2

Offset: 1

Peter Bala, Dec 13 2012

The Engel expansion of a positive real number x is the unique nondecreasing sequence {e(1), e(2), e(3), ...} of positive integers such that x = 1/e(1) + 1/(e(1)*e(2)) + 1/(e(1)*e(2)*e(3)) + .... The terms in the Engel expansion of x are obtained from the iterates of the mapping g(x) := x*(1 + floor(1/x)) - 1 by means of the formula e(n) = 1 + floor(1/g^(n)(x)).
Let h(x) = floor(1/x)*g(x). This is called the harmonic sawtooth map by Crowley. Then the modified Engel expansion of a real number 0 < x <= 1 is a sequence {a(1), a(2), a(3), ...} of positive integers such that x = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + ... whose terms are obtained from the iterates of the harmonic sawtooth map h(x) by the formulas a(1) = 1 + floor(1/x) and, for n >= 2, a(n) = floor(1/h^(n-2)(x))*{1 + floor(1/h^(n-1)(x))}. Here h^(n)(x) = h(h^(n-1)(x)) denotes the n-th iterate of the map h(x), with the convention that h^(0)(x) = x. For further details see the Bala link.
When the real number x > 1 with, say, floor(x) = m, the modified Engel expansion of x is found by first calculating the modified Engel expansion of x - m and then prepending a sequence of m 1's to this.

Peter Bala, A modified Engel expansion
Wikipedia, Engel Expansion

Cf. A006784, A220335, A220336, A220337, A220338, A220394, A220395, A220396, A220397, A220398.

Let x = Pi - 3. Then a(1) = a(2) = a(3) = 1, a(4) = ceiling(1/x) and, for n >= 1, a(n+4) = floor(1/h^(n-1)(x))*ceiling(1/h^(n)(x)).

Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion Pi = Sum_{n>=1} 1/P(n) = 1/1 + 1/1 + 1/1 + 1/8 + 1/(8*14) + 1/(8*14*2) + 1/(8*14*2*2) + .... For n >= 4, the error made in truncating this series to n terms is less than the n-th term.

A228929 Optimal ascending continued fraction expansion of Pi - 3.

Original entry on oeis.org

7, -113, 4739, -46804, 134370, -614063, 1669512, -15474114, -86232481, 1080357006, -8574121305, -24144614592, 133884333083, -2239330253016, -6347915250018, 14541933941298, -42301908155404, -298013673554972, 5177473084279656, -46709468571434452, 1201667304102142095, -68508286025632748778, 850084640720511629243, -2458418086834560217354

Offset: 1

Giovanni Artico, Sep 08 2013

sign

Definition of the expansion: for a positive real number x, there is always a unique sequence of signed integers with increasing absolute value |a(i)|>|a(i-1)| such that x =floor(x)+ 1/a(1) + 1/a(1)/a(2) + 1/a(1)/a(2)/a(3) + 1/a(1)/a(2)/a(3)/a(4) ... or equivalently x=floor(x)+1/a(1)*(1+1/a(2)*(1+1/a(3)*(1+1/a(4)*(1+...)))) giving the fastest converging series with this representation. This formula can be represented as a regular ascending continued fraction. The expansion is similar to Engel and Pierce expansions, but the sign of the terms is not predefined and determined by the algorithm for optimizing the convergence.

For x rational number the expansion has a finite number of terms, for x irrational an infinite number. Empirically the sequence doesn't show any evident regularity except in some interesting cases.

			Pi = 3 + 1/7*(1 - 1/113*(1 + 1/4739*(1 - 1/46804*(1 + 1/134370*(1 - 1/614063*(1 + 1/1669512*(1 + ...))))))).

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A006784, A015884, A228930, A228931, A228932, A228933, A228934.

ArticoExp(x, n) := VECTOR(ROUND(1, ABS(k))*SIGN(k), k, ITERATES(ROUND(1, ABS(u))*ABS(u) - 1, u, MOD(x), n))
Precision:=Mixed
PrecisionDigits:=10000
ArticoExp(PI,20)

# Slow procedure valid for every number
ArticoExp := proc (n, q::posint)::list; local L, i, z; Digits := 50000; L := []; z := n-floor(n); for i to q+1 do if z = 0 then break end if; L := [op(L), round(1/abs(z))*sign(evalf(z))]; z := abs(z)*round(1/abs(z))-1 end do; return L end proc
# Fast procedure, not suited for rational numbers
ArticoExp := proc (n, q::posint)::list; local L, i, z; Digits := 50000; L := []; z := frac(evalf(n)); for i to q+1 do if z = 0 then break end if; L := [op(L), round(1/abs(z))*sign(z)]; z := abs(z)*round(1/abs(z))-1 end do; return L end proc
# List the first 20 terms of the expansion of Pi
ArticoExp(Pi,20)

ArticoExp[x_, n_] :=  Round[1/#] & /@ NestList[Round[1/Abs[#]]*Abs[#] - 1 &, FractionalPart[x], n]; Block[{$MaxExtraPrecision = 50000}, ArticoExp[Pi, 20]]

Given a positive real number x, let z(0)=x-floor(x) and z(k+1)=abs(z(k))*round(1/abs(z(k)))-1 ; then a(n)=sign(z(n))*round(1/abs(z(n))) for n>0.

A028259 Engel expansion of the golden ratio, (1 + sqrt(5))/2 = 1.61803... .

Original entry on oeis.org

1, 2, 5, 6, 13, 16, 16, 38, 48, 58, 104, 177, 263, 332, 389, 4102, 4575, 5081, 9962, 18316, 86613, 233239, 342534, 964372, 1452850, 7037119, 7339713, 8270361, 12855437, 15900982, 19211148, 1365302354, 1565752087, 1731612283

Offset: 1

Naoki Sato (naoki(AT)math.toronto.edu), Dec 11 1999

Cf. A006784 for definition of Engel expansion.
(sqrt(5) - 1)/2 = 1/(golden ratio) has the same Engel expansion starting with a(2). - G. C. Greubel, Oct 16 2016
Each Engel expansion for a constant x (x > 0) starts with floor(x) 1's. After that, the mean growth of the terms is by a factor e for most constants x, i.e., the order of magnitude of the n-th entry is exp(n-floor(c)), for n large enough. This comment is similar to the fact that for the continued fraction terms of most constants, the geometric mean of those terms equals the Khintchine constant for n large enough. Moreover, note that for the golden section, all continued fraction terms are 1 and thus do not obey the Gauss-Kuzmin distribution which leads to the Khintchine constant (i.e., the Khintchine measure for the golden ratio is 1), but the Engel expansion does obey the statistic behavior for most constants. - A.H.M. Smeets, Aug 24 2018

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

A.H.M. Smeets, Table of n, a(n) for n = 1..2406 (Terms 1 through 300 from T. D. Noe, 301 through 698 from Simon Plouffe, and 699 through 1500 from G. C. Greubel)
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Golden Ratio
Index entries for sequences related to Engel expansions

Cf. A001622, A220398.

EngelExp[ A_, n_ ] := Join[ Array[ 1&, Floor[ A ] ], First@Transpose@NestList[ {Ceiling[ 1/Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ] ], Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ]}&, {Ceiling[ 1/(A-Floor[ A ]) ], A-Floor[ A ]}, n-1 ] ]

j = 0
while j<3100000:
# to obtain n correct Engel expansion terms about n^2/2 continued fraction steps are needed; 3100000 is a safe bound
    if j == 0:
        p0, q0 = 1, 1
    elif j == 1:
        p1, q1 = p0+1, 1
    else:
        p0, p1 = p1, p1+p0
        q0, q1 = q1, q1+q0
    j = j+1
i = 0
while i < 2410:
    i = i+1
    a = q0//p0+1
    print(i, a)
    p0 = a*p0-q0
# A.H.M. Smeets, Aug 24 2018

Limit_{n -> oo} log(a(n + floor(golden ratio)))/n = 1.

Corrected and extended by Vladeta Jovovic, Aug 16 2001

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